EDIT - I did it! I used Handbrake to convert the .mp4 into an .mp4 (yeeah) but under the "video" tab selected "MPEG-4 (FFmpeg)" instead of the default one. So I have no clue what that means, but it allowed me to use Movie Maker to crop and extract the audio, which has uploaded and plays on my phone, which is the whole point of this ridiculous exercise.
So Movie Maker does indeed strip audio (woohoo!)... BUT. It only opens certain .mp4 files. So apparently an .mp4 is not always an .mp4.
The only difference I can see between the two types of .mp4 files is that in windows explorer, the ones that come up with a thumbnail will open in Movie Maker; and the ones that don't have a thumbnail won't. Coincidence? Critical clue? Who knows!